Computed tomography device and method using circular-pixel position-adaptive interpolation

ABSTRACT

A method of computed-tomography and a computed-tomography apparatus where a interpolation kernel width is adaptively determined as a function of the distance from the x-ray source to the reconstruction pixel. The width of the kernel is the projection of the reconstruction pixel on the detector. The method can be implemented in the channel direction. The method can also be implemented in the segment direction, or in the channel and segment directions at the same time. Backprojection is performed using the adaptive kernel width and may by used with helical and circular scanning, and with cone-beam or fan beam x-ray CT.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to helical and circular x-ray computedtomographic (CT) imaging, and in particular to CT imaging with adaptiveinterpolation.

2. Discussion of the Background

There are several methods of backprojection. A common method ispixel-driven backprojection illustrated in FIG. 1. A ray 11 is drawnfrom the source 10 through the center of the reconstruction pixel at(i,j) to the detector array 12. Ray 11 intersects detector 12 atfloating point channel index Ch_(i,j) on channel c_(k). The valueassigned to the pixel is a linear interpolation of the values of thedetector the ray intersects and the adjacent elements:

V(i,j)=w _(k−1) ·D(c _(k−1))+w _(k) ·D(c _(k))+w _(k+1) ·D(c_(k+1))  (1)

where c_(k) is the channel of intersection and w_(k−1)+w_(k)+w_(k+1)=1.For typical linear interpolation, the interpolation kernel width ΔCh isfixed at one channel and only two detector elements are used (eitherw_(k−1) or w_(k+1) is zero). A nearest neighbor implementation hasw_(k+1)=w_(k−1)=0 and w_(k)=1. Another pixel-driven method adjusts theweights such that interpolation kernel locations on the detector arealways adjacent, as described in U.S. Pat. No. 6,724,856.

Pixel-driven methods are easy to implement. However, the disadvantage isthat actual projection of the pixel onto the detector array is not usedto compute V(i,j). For large pixel sizes, the projection of the pixelonto the array will be larger than one channel, meaning that too fewdetector elements contribute to V(i,j), and the signal-to-noise ratio(SNR) will be poorer. For small pixel sizes the projection of the pixelonto the array is smaller than one channel, causing a loss of lateralresolution.

Another backprojection method is ray-driven backprojection. Thistechnique is shown in FIG. 2 and described in Barrett, Harrison andSwindell, William, “Radiological Imaging: The Theory of Image Formation,Detection, and Processing,” New York: Academic Press, 1981, pp. 421-422.In this case the backprojection ray 21 is drawn from the source 20 tothe centers (indicated as c) of detector elements in the detector array22, and the contribution weight w_(ijc) of the detector element topixels in the ray path is determined from the length of the ray withinthe pixel, as follows:

V(i,j)=V(i,j)+w _(ijc) ·D(c)  (2)

Other ray driven methods perform linear interpolation between two pixelsfor each row or column intersected by the projection line, as describedin Joseph P, “An improved algorithm for reprojecting rays through pixelimages,” IEEE Trans. Med. Imaging 1, pp. 192-196 (1982).

A third technique, distance-driven backprojection, maps the boundariesof the pixels and the detector elements onto a common axis (referred toas centering), and the amount of overlap is used as the backprojectionweight, as shown in FIG. 3. Rays 31 from source 30 strike array 32. SeeU.S. Pat. No. 7,227,982 and De Man, Bruno, and Basu, Samit,“Distance-driven projection and backprojection in three dimensions,”Phys Med Biol 49, pp. 2463-2475 (2004).

The pixel 33 is assumed to be square with a half-width δ, and the edgesof the square pixel are in the x-ray beam 31 from source 30 areprojected to points x_(pc1) and x_(pc2) on the common centering x axis.Similarly, the boundaries of detectors c are projected to positions xccon the x axis. In FIG. 3, channel boundaries c_(s) to c_(e+1) areprojected to positions xcc_(cs) to xcc_(ce+1) on the centering axis. Theinterpolation kernel width is the overlap of x_(pc1) to x_(pc2) withxcc_(cs) to xcc_(ce+1):

$\begin{matrix}{{V\left( {i,j} \right)} = {\frac{f_{iscc} \cdot {D\left\lbrack c_{s} \right\rbrack}}{\left( {{ycc}_{ks} - {ycc}_{{ks} + 1}} \right)} + \frac{f_{iecc} \cdot {D\left\lbrack c_{e} \right\rbrack}}{\left( {{ycc}_{ke} - {ycc}_{{ke} + 1}} \right)} + {\sum\limits_{c^{\prime} = {c_{s} + 1}}^{c_{e} - 1}{D\left\lbrack c^{\prime} \right\rbrack}}}} & (3)\end{matrix}$

The advantage of this method is that the projection of the pixel ontothe detector is used to determine the interpolation kernel width,yielding improved SNR at large pixel sizes and improved resolution atsmaller pixel sizes. The disadvantage to this method is that both thechannels and pixels must be projected onto a centering axis (typicallythe y or x axis), which makes processing more complex. Furthermore, theresulting interpolation kernel width is a function of view.

SUMMARY OF THE INVENTION

The present invention, in one embodiment, is directed to acomputed-tomography method including determining a channel positionlocated on an x-ray detector using a linear path from an x-ray sourcethrough a backprojection pixel, and determining an interpolation kernelwidth using a distance from the source to the backprojection pixel.

In another embodiment, the present invention is directed to an apparatushaving an x-ray source, an x-ray detector having channels and segments,a signal processing unit connected to receive data collected thedetector, and configured to determine a channel position on the detectorlocated on a linear path from the x-ray source through a backprojectionpixel and determine an interpolation kernel width using a distance fromthe source to the backprojection pixel.

The present invention may also be directed to a computer-readable mediumcontaining instructions, wherein the instructions, when executed by acomputer, perform a method including determining a channel positionlocated on a linear path from an x-ray source through a backprojectionpixel, and determining an interpolation kernel width using a distancefrom the source to the backprojection pixel.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is a diagram illustrating conventional pixel-drivenbackprojection;

FIG. 2 is a diagram illustrating conventional ray-driven backprojection;

FIG. 3 is a diagram illustrating conventional distance-drivenbackprojection;

FIG. 4 is a diagram of a system according to the invention;

FIG. 5 is a diagram illustrating possible axial spatial orientations foradjacent views with resulting interpolated views;

FIG. 6 is a diagram illustrating backprojection according to theinvention;

FIG. 7 is a diagram illustrating calculation of the backprojectionvalue;

FIG. 8 is a graph illustrating backprojection according to the inventionfor different field of views;

FIG. 9 is a graph illustrating a comparison between backprojectionaccording to the invention and distance-driven backprojection atdifferent field of views;

FIGS. 10A-10E are graphs illustrating a comparison betweenbackprojection according to the invention and distance-drivenbackprojection at different field of views; and

FIG. 11 is a diagram illustrating a second embodiment of the invention.

DETAILED DESCRIPTION

FIG. 4 shows an x-ray computed tomographic imaging device according tothe present invention. The projection data measurement systemconstituted by gantry 1 accommodates an x-ray source 3 that generates acone-beam of x-ray flux approximately cone-shaped, and a two-dimensionalarray type x-ray detector 5 consisting of a plurality of detectorelements arranged in two-dimensional fashion, i.e., a plurality ofelements arranged in one dimension stacked in a plurality of rows. X-raysource 3 and two-dimensional array type x-ray detector 5 are installedon a rotating ring 2 in facing opposite sides of a subject, who is laidon a sliding sheet of a bed 6. Two-dimensional array type x-ray detector5 is mounted on rotating ring 2. Each detector element will correspondwith one channel. X-rays from x-ray source 3 are directed on to subjectthrough an x-ray filter 4. X-rays that have passed through the subjectare detected as an electrical signal by two-dimensional array type x-raydetector 5.

X-ray controller 8 supplies a trigger signal to high voltage generator7. High voltage generator 7 applies high voltage to x-ray source 3 withthe timing with which the trigger signal is received. This causes x-raysto be emitted from x-ray source 3. Gantry/bed controller 9 synchronouslycontrols the revolution of rotating ring 2 of gantry 1 and the slidingof the sliding sheet of bed 6. System controller 10 constitutes thecontrol center of the entire system and controls x-ray controller 8 andgantry/bed controller 9 such that, as seen from the subject, x-raysource 3 executes so-called helical scanning, in which it moves along ahelical path. Specifically, rotating ring 2 is continuously rotated withfixed angular speed while the sliding plate is displaced with fixedspeed, and x-rays are emitted continuously or intermittently at fixedangular intervals from x-ray source 3. The source may also be scannedcircularly.

The output signal of two-dimensional array type x-ray detector 5 isamplified by a data collection unit 11 for each channel and converted toa digital signal, to produce projection data. The projection data outputfrom data collection unit 11 is fed to processing unit 12. Processingunit 12 performs various processing using the projection data. Unit 12performs interpolation, backprojection and reconstruction, as describedin more detail below. Unit 12 determines backprojection data reflectingthe x-ray absorption in each voxel. In the helical scanning system usinga cone-beam of x-rays, the imaging region (effective field of view) isof cylindrical shape of radius o) centered on the axis of revolution.Unit 12 defines a plurality of voxels (three-dimensional pixels) in thisimaging region, and finds the backprojection data for each voxel. Thethree-dimensional image data or tomographic image data compiled by usingthis backprojection data is sent to display device 14, where it isdisplayed visually as a three-dimensional image or tomographic image.

The basic geometry of the backprojection according to the invention isillustrated in FIG. 5. The backprojection according to the invention mayby used with helical and circular scanning, and with cone-beam or fanbeam x-ray CT. It incorporates the pixel-driven attribute of tracing aline 51 from the source 50 through the center of the pixel 53 (havingradius P) to the detector 52 to determine the backprojection positionCh_(i,j), and adaptively determines the interpolation kernel width ΔChas a function of the source to pixel distance L assuming the pixel has acircular shape. The backprojection according to the invention does notrequire projection of pixels and detector boundaries onto a centeringaxis, yet retains the advantage of distance-based backprojection; i.e.,improved signal-to-noise ratio (SNR) at large pixel sizes and improvedresolution at smaller pixel sizes.

For a 2-D implementation, the adaptive kernel is applied only in thechannel direction, and bilinear interpolation in the segment direction.The backprojection according to the invention in the channel directionis shown in more detail in FIG. 6, with segments omitted for clarity.The pixel is assumed to have a circular shape with center at positionx_(i,j), y_(i,e) and radius P:

$\begin{matrix}{P = \frac{F\; O\; V}{2 \cdot {MATRIX}}} & (4)\end{matrix}$

where FOV is the reconstruction field of view and MATRIX is thereconstructed image matrix size. The floating point channel positionCh_(i,j) (in units of channels) is determined in the same manner as fora pixel-driven method:

$\begin{matrix}{{Ch}_{i,j,k} = {\frac{1}{\Delta \; \gamma}{\tan^{- 1}\left\lbrack \frac{{x_{i,j}\cos \; \beta_{k}} + {y_{i,j}\sin \; \beta_{k}}}{R + {x_{i,j}\sin \; \beta_{k}} - {y_{i,j}\cos \; \beta_{k}}} \right\rbrack}}} & (5)\end{matrix}$

where Δγ the angular channel width, β_(k) is the view angle at view k,and R is the source to isocenter distance, where the isocenter is thegantry rotation axis. The projection of the circular pixel determinesthe adaptive channel interpolation kernel width (in units of channels),which is given by:

$\begin{matrix}{{{\Delta \; {Ch}_{i,j,k}} = \frac{2\; P}{\Delta \; {\gamma \cdot L_{i,j,k}}}}{where}} & (6) \\{{L_{i,j,k}\left( {x_{i,j},y_{i,j},\beta_{k}} \right)} = \sqrt{\left( {{R\; \sin \; \beta_{k}} + x_{i,j}} \right)^{2} + \left( {{R\; \cos \; \beta_{k}} - y_{i,j}} \right)^{2}}} & (7)\end{matrix}$

Calculation of the channel interpolated backprojection value isillustrated in FIG. 7 Here, c is an integer and represents the channelnumber that a backprojection ray intersects. In this example, it isassumed that the indexing of the channels is such that the first channelis located at c=0, and that a ray intersecting channel 0 exactly in thecenter will have floating point position 0.5 (i.e., possible range ofpositions for channel 0 is 0.0 to 0.9999 . . . ) Other indexing schemesmay have the center of channel 0 at 0.0, with a possible range ofpositions from −0.5 to +0.49999 . . . . If a different indexing schemeis used, Eq. 5 will be accordingly altered by adding a shift amount.

$\begin{matrix}{{PDI}_{{CH}_{i,j,k}} = \left\{ \begin{matrix}{{PD}(c)} & {{{if}\mspace{14mu} c_{s}} = {{c\mspace{14mu} {and}\mspace{14mu} c_{e}} = c}} \\{\frac{1}{\Delta \; {Ch}_{i,j,k}}\begin{pmatrix}{{\delta_{s} \cdot {{PD}\left( c_{s} \right)}} + {\delta_{e} \cdot}} \\{{{PD}\left( c_{e} \right)} + {\sum\limits_{c^{\prime} = {c_{s} + 1}}^{c_{e} - 1}{{PD}\left( c^{\prime} \right)}}}\end{pmatrix}} & {otherwise}\end{matrix} \right.} & (8)\end{matrix}$

where PD( ) is raw projection data, PDI is interpolated value ofprojection data, c_(s), c, and c_(e) are integer starting, center, andending channel indexes and

$\begin{matrix}{\lambda = \frac{\Delta \; {Ch}_{i,j,k}}{2}} & (9) \\{f_{s} = {{Ch}_{i,j} - \lambda}} & (10) \\{f_{e} = {{Ch}_{i,j} + \lambda}} & (11) \\{c_{s} = {{int}\left( f_{s} \right)}} & (12) \\{c_{e} = {{int}\left( f_{e} \right)}} & (13) \\{\delta_{s} = {c_{s + 1} - f_{s}}} & (14) \\{\delta_{e} = {f_{e} - c_{e}}} & (15)\end{matrix}$

FIG. 8 shows an example of the interpolation kernel ΔCh as a function ofL for various FOV sizes. For standard pixel-driven backprojection, ΔChis equal to 1.0. Smaller FOVs are typically used to visualize smallerstructures, where a higher resolution is desirable. For small FOVs of240 and less, ΔCh<1, which provides a better resolution. For largerFOVs, resolution is usually not the issue. Here ΔCh>1, which provides abetter signal-to-noise ratio, since more channels are used in theinterpolation as well as better aliasing artifact.

FIG. 9 shows a comparison of ΔCh between the present invention (solidlines) and distance driven backprojection (DDBPJ, dashed lines) atextremes of view angle β=0° and β=45° (which are extremes for DDBPJ forthe centering process). These are shown as a function of angular pixelposition φ at radii of various FOVs. Overall, ΔCh for the presentinvention is somewhat larger than DDBPJ for a given FOV and pixelposition, but will have similar image quality.

FIGS. 8 and 9 show that the method according to the invention canprovide similar performance in terms of images compared to theconventional methods, while the method according to the invention.

In the first embodiment, the method is implemented in the channeldirection. In an alternate embodiment, the method can be implemented inthe segment direction by assuming the voxel is circular in z andcalculating the projection of the circle across the segments in the zdirection, as shown in FIG. 10. Q is the pixel radius in z and is equalto the reconstruction slice spacing (though in other embodiments it canbe calculated differently). FIGS. 10A-E show results for FOVs of 500,400, 320, 240 and 180, respectively. The solid lines represent thepresent invention and the dashed lines show, for comparison, distancedriven backprojection.

The floating point segment position is determined exactly the same as itis for the pixel-driven method:

$\begin{matrix}{z_{seg} = \frac{R \cdot z_{p}}{L}} & (16)\end{matrix}$

where z_(seg) is the distance from the then center of the detector tothe backprojection segment location and is z_(p) is the z distance ofthe reconstruction pixel from the source-detector plane at view k. Forcircular scanning this is independent of view and for helical it is afunction of view. The floating point segment position Seg is given by

$\begin{matrix}{{Seg} = \frac{z_{seg}}{w}} & (17)\end{matrix}$

where w is the segment width at isocenter.

The adaptive segment interpolation kernel width ΔSeg (in units ofsegments) given by:

$\begin{matrix}{{{\Delta \; {Seg}} = \frac{2\; \sigma_{seg}}{w}}{where}} & (18) \\{\sigma_{seg} = {{Q \cdot \cos}\; \theta \frac{d_{seg}}{d_{p}}}} & (19) \\{d_{seg} = \sqrt{R^{2} + z_{seg}^{2}}} & (20) \\{d_{p} = \sqrt{L^{2} + z_{p}^{2}}} & (21) \\{\theta = {\tan^{- 1}\frac{z_{p}}{L}}} & (22)\end{matrix}$

Calculation of the segment backprojection value is calculated in thesame manner as for the channels as in FIG. 7 and Eq. 8, substitutingsegments for channels.

In a third embodiment, the method can be implemented for both thechannel and segment directions at the same time. This is the 3-D case,where the circular pixel is now a spherical voxel with radius S.Equations (6-15) are used to calculate channel direction projection withP=S and equations (16-22) for segment direction, with Q=S. Thisembodiment can be used for full cone-beam detectors.

The invention may also be embodied in the form a computer-readablemedium containing a stored program to cause a computer to carry out thevarious operations and functions described above.

Numerous other modifications and variations of the present invention arepossible in light of the above teachings. This document and equationshave been developed for a curved detector array. For example, a flat orother detector array shape can be implemented. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described herein.

1. A computed-tomography method, comprising: determining a channelposition located on an x-ray detector using a linear path from an x-raysource through a backprojection pixel; and determining an interpolationkernel width using a distance from said source to said backprojectionpixel.
 2. A method as recited in claim 1, comprising: determining saidkernel with as a projection of said backprojection pixel onto saiddetector.
 3. A method as recited in claim 1, comprising: determining aradius P of said pixel having a circular shape with a center at positionx_(i,j), y_(i,j) as $P = \frac{F\; O\; V}{2 \cdot {MATRIX}}$ whereFOV is a reconstruction field of view and MATRIX is a reconstructedimage matrix size; determining said position Ch_(i,j) as:${Ch}_{i,j,k} = {\frac{1}{\Delta \; \gamma}{\tan^{- 1}\left\lbrack \frac{{x_{i,j}\cos \; \beta_{k}} + {y_{i,j}\sin \; \beta_{k}}}{R + {x_{i,j}\sin \; \beta_{k}} - {y_{i,j}\cos \; \beta_{k}}} \right\rbrack}}$where Δγ an angular channel width, β_(k) is a view angle at view k, andR is a distance from said x-ray source to isocenter distance; anddetermining said kernel width ΔCh_(i,j) as:${\Delta \; {Ch}_{i,j,k}} = \frac{2\; P}{\Delta \; {\gamma \cdot L_{i,j,k}}}$where${L_{i,j,k}\left( {x_{i,j},y_{i,j},\beta_{k}} \right)} = {\sqrt{\left( {{R\; \sin \; \beta_{k}} + x_{i,j}} \right)^{2} + \left( {{R\; \cos \; \beta_{k}} - y_{i,j}} \right)^{2}}.}$4. A method as recited in claim 3, comprising: calculating abackprojection value PDI as:${PDI}_{{CH}_{i,j,k}} = \left\{ \begin{matrix}{{PD}(c)} & {{{if}\mspace{14mu} c_{s}} = {{c\mspace{14mu} {and}\mspace{14mu} c_{e}} = c}} \\{\frac{1}{\Delta \; {Ch}_{i,j,k}}\begin{pmatrix}{{\delta_{s} \cdot {{PD}\left( c_{s} \right)}} + {\delta_{e} \cdot}} \\{{{PD}\left( c_{e} \right)} + {\sum\limits_{c^{\prime} = {c_{s} + 1}}^{c_{e} - 1}{{PD}\left( c^{\prime} \right)}}}\end{pmatrix}} & {otherwise}\end{matrix} \right.$ where PD( ) is raw projection data, c_(s), c, andc_(e) are integer starting, center, and ending channel indexes of adetector, and $\lambda = \frac{\Delta \; {Ch}_{i,j,k}}{2}$f_(s) = Ch_(i, j) − λ f_(e) = Ch_(i, j) + λ c_(s) = int(f_(s))c_(e) = int(f_(e)) δ_(s) = c_(s + 1) − f_(s), andδ_(e) = f_(e) − c_(e).
 5. A method as recited in claim 1, comprising:determining a backprojection value inversely proportional to said kernelwidth.
 6. A computed-tomography method for use in a system having anx-ray source and an x-ray detector with segments, comprising:determining a segment position of said x-ray detector; and determining asegment kernel width as a function of a first distance from a center ofsaid detector to a backprojection segment location and a second distanceof a reconstruction pixel from a plane defined by the source anddetector.
 7. A method as recited in claim 6, comprising: determiningsaid segment position Seg as ${Seg} = \frac{z_{seg}}{w}$ where w is asegment width at isocenter of said detector and said source and$z_{seg} = \frac{R \cdot z_{p}}{L}$ where said reconstruction pixel isat a position x_(i,j), y_(i,j), R is a distance between said source andsaid isocenter, β_(k) is a view angle at a view k, andL _(i,j,k)(x _(i,j) ,y _(i,j),β_(k))=√{square root over ((R sin β_(k) +x_(i,j))²+(R cos β_(k) −y _(i,j))²)}{square root over ((R sin β_(k) +x_(i,j))²+(R cos β_(k) −y _(i,j))²)}; determining said kernel width as${\Delta \; {Seg}} = \frac{2\; \sigma_{seg}}{w}$ where$\sigma_{seg} = {{Q \cdot \cos}\; \theta \frac{d_{seg}}{d_{p}}}$$d_{seg} = \sqrt{R^{2} + z_{seg}^{2}}$$d_{p} = \sqrt{L^{2} + z_{p}^{2}}$$\theta = {\tan^{- 1}{\frac{z_{p}}{L}.}}$
 8. An apparatus, comprising:an x-ray source; an x-ray detector having channels and segments; asignal processing unit connected to receive data collected saiddetector, and configured to determine a channel position on saiddetector located on a linear path from said x-ray source through abackprojection pixel and determine an interpolation kernel width using adistance from said source to said backprojection pixel.
 9. An apparatusas recited in claim 8, wherein said signal processing unit is configuredto determine a backprojection value inversely proportional to saidkernel width.
 10. An apparatus as recited in claim 8, wherein: saiddetector comprises segments; and said signal processing unit isconfigured to determine a segment kernel width as a function of a firstdistance from a center of said detector to a backprojection segmentlocation and a second distance of a reconstruction pixel from a planedefined by the source and detector.
 11. An apparatus as recited in claim10, wherein said signal processing unit is configured to determine abackprojection value inversely proportional to said kernel width.
 12. Anapparatus as recited in claim 10, wherein said signal processing unit isconfigured to determine said kernel width as a projection of saidbackprojection pixel onto said detector.
 13. A computer-readable mediumcontaining instructions, wherein the instructions, when executed by acomputer, perform a method comprising: determining a channel positionlocated on a linear path from an x-ray source through a backprojectionpixel; and determining an interpolation kernel width using a distancefrom said source to said backprojection pixel.
 14. A computer-readablemedium as recited in claim 13, wherein said method further comprising:determining a radius P of said pixel having a circular shape with acenter at position x_(i,j), y_(i,j) as$P = \frac{F\; O\; V}{2 \cdot {MATRIX}}$ where FOV is areconstruction field of view and MATRIX is a reconstructed image matrixsize; determining said position Ch_(i,j) as:${Ch}_{i,j,k} = {\frac{1}{\Delta \; \gamma}{\tan^{- 1}\left\lbrack \frac{{x_{i,j}\cos \; \beta_{k}} + {y_{i,j}\sin \; \beta_{k}}}{R + {x_{i,j}\sin \; \beta_{k}} - {y_{i,j}\cos \; \beta_{k}}} \right\rbrack}}$where Δγ is an angular channel width, β_(k) is a view angle at view k,and R is a distance from said x-ray source to isocenter distance; anddetermining said kernel width ΔCh_(i,j) as:${\Delta \; {Ch}_{i,j,k}} = \frac{2\; P}{\Delta \; {\gamma \cdot L_{i,j,k}}}$where${L_{i,j,k}\left( {x_{i,j},y_{i,j},\beta_{k}} \right)} = {\sqrt{\left( {{R\; \sin \; \beta_{k}} + x_{i,j}} \right)^{2} + \left( {{R\; \cos \; \beta_{k}} - y_{i,j}} \right)^{2}}.}$15. A medium as recited in claim 13, wherein said method furthercomprises: calculating a backprojection value PDI as:${PDI}_{{CH}_{i,j,k}} = \left\{ \begin{matrix}{{PD}(c)} & {{{if}\mspace{14mu} c_{s}} = {{c\mspace{14mu} {and}\mspace{14mu} c_{e}} = c}} \\{\frac{1}{\Delta \; {Ch}_{i,j,k}}\begin{pmatrix}{{\delta_{s} \cdot {{PD}\left( c_{s} \right)}} + {\delta_{e} \cdot}} \\{{{PD}\left( c_{e} \right)} + {\sum\limits_{c^{\prime} = {c_{s} + 1}}^{c_{e} - 1}{{PD}\left( c^{\prime} \right)}}}\end{pmatrix}} & {otherwise}\end{matrix} \right.$ where PD( ) is raw projection data, c_(s), c, andc_(e) are integer starting, center, and ending channel indexes of adetector, and $\lambda = \frac{\Delta \; {Ch}_{i,j,k}}{2}$f_(s) = Ch_(i, j) − λ f_(e) = Ch_(i, j) + λ c_(s) = int(f_(s))c_(e) = int(f_(e)) δ_(s) = c_(s + 1) − f_(s), andδ_(e) = f_(e) − c_(e).
 16. A medium as recited in claim 13, wherein saidmethod comprises: determining a backprojection value inverselyproportional to said kernel width.
 17. A computer-readable mediumcontaining instructions, wherein the instructions, when executed by acomputer, perform a method comprising: determining a segment position ofan x-ray detector; and determining a segment kernel width as a functionof a first distance from a center of said detector to a backprojectionsegment location and a second distance of a reconstruction pixel from aplane defined by the source and detector.
 18. A medium as recited inclaim 16, comprising: determining said segment position Seg as${Seg} = \frac{z_{seg}}{w}$ where w is a segment width at isocenter ofsaid detector and said source and $z_{seg} = \frac{R \cdot z_{p}}{L}$where said reconstruction pixel is at a position x_(i,j), y_(i,j), R isa distance between said source and said isocenter, β_(k) is a view angleat a view k, andL _(i,j,k)(x _(i,j),y_(i,j),β_(k))=√{square root over ((R sin β_(k) +x_(i,j))²+(R cos β_(k) −y _(i,j))²)}{square root over ((R sin β_(k) +x_(i,j))²+(R cos β_(k) −y _(i,j))²)}; determining said kernel width as${\Delta \; {Seg}} = \frac{2\; \sigma_{seg}}{w}$ where$\sigma_{seg} = {{Q \cdot \cos}\; \theta \frac{d_{seg}}{d_{p}}}$$d_{seg} = \sqrt{R^{2} + z_{seg}^{2}}$$d_{p} = \sqrt{L^{2} + z_{p}^{2}}$$\theta = {\tan^{- 1}{\frac{z_{p}}{L}.}}$